K3-component decompositions for nested Debarre–Voisin varieties

Let $Y\subset\operatorname{Gr}(3,V_{10})$ be a very general hyperplane section, and let $\mathsf{A}$ be the K3 subcategory of $\operatorname{D}^b(Y)$ obtained as the semiorthogonal complement of $108$ exceptional objects. Let $Y_1$, $Y_2$, $T$, and $P$ be the nested varieties associated with $Y$. Nested K3-decomposition conjecture. Up to equivalence, the following semiorthogonal decompositions should hold:

$$\operatorname{D}^b(Y_1)=\langle\mathsf{A},48\text{ exceptional objects}\rangle,$$ $$\operatorname{D}^b(Y_2)=\langle\mathsf{A},24\text{ exceptional objects}\rangle,$$ $$\operatorname{D}^b(T)=\langle\mathsf{A},\mathsf{A},\mathsf{A},9\text{ exceptional objects}\rangle,$$ $$\operatorname{D}^b(P)=\langle\mathsf{A},\mathsf{A},\mathsf{A},4\text{ exceptional objects}\rangle.$$

In particular, $Y_1$ and $Y_2$ should be of derived pure K3 type, whereas $P$ and $T$ should be of derived non-pure K3 type. The paper presents evidence but explicitly says that these decompositions are not proved.